Non-commutative geometry of finite groups

Abstract

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right-and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of 'extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a 'dual connection' which exists on the dual bimodule (as defined in this work).

Noncommutative Physics on Lie Algebras, (ℤ2) n Lattices and Clifford Algebras

“…There is a standard construction for the wedge product of basic forms as well. This set up is an immediate corollary of the analysis of [5] but has been emphasised by many authors, such as [20] or more recently [21] [22]. Moreover, one can take a metric δ a,b −1 in the {e a } basis, which leads to a canonical Hodge operation if the exterior algebra is finite dimensional.…”

“…This is consistent with our view that the above predictions have nothing to do with gravity, it is an independent effect. Thus, the curvature in the Maxwell theory of a gauge field A = A µ dx µ is (20) and its components have the usual antisymmetric form because the basic forms anticommute as usual. Because the Hodge operations on them are also as usual when we keep all differentials to the right, and because the partial derivatives commute, the Maxwell operator d d on 1-forms has the same form as the usual one, namely the scalar wave operator as above if we take A in Lorentz gauge ∂ µ A µ = 0.…”

Non-commutative geometry and physics: a review of selected recent results

“…This method is based on the (noncommutative) differential geometry of finite groups, studied in ref.s [45,46,48,47]. The general theory is applied to the simplest possible finite group, i.e.…”

“…ad(h)g ≡ hgh −1 . Then bicovariant calculi are in 1-1 correspondence with unions of conjugacy classes (different from {e}) [45]: if θ g is set to zero, one must set to zero all the θ ad(h)g , ∀h ∈ G corresponding to the whole conjugation class of g.…”

Riemannian geometry of bicovariant group lattices

“…A group lattice, which is determined by a discrete group G and a finite subset S (which does not contain the unit element e) naturally defines a first-order differential calculus (which extends to higher orders) over the algebra A of functions on G. If S generates G, then bicovariance of the group lattice (G, S) is equivalent to bicovariance of the first-order differential calculus in the sense of Ref. 2. "Riemannian geometry" of discrete groups in the context of noncommutative geometry has already been considered in several publications [3,4,5]. The present approach differs from these in particular by introducing a metric tensor as an element of a left-covariant tensor product of the space of 1-forms with itself.…”